Problem Set 9—Double Integrals

Purpose

This problem set develops your ability to evaluate and use integrals over 2-dimensional regions.

Background

This exercise is based on material in sections 15.1 through 15.3 of our textbook. We will discuss this material in class between November 6 and November 11.

Activity

Solve each of the following problems:

Problem 1

Exercise 26 in section 15.1 of our textbook (find the volume of the region between 16 - x² - y² and the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 2 in the xy plane).

Problem 2

Exercise 34 in section 15.1 of our textbook (evaluate the integral from 0 to 1 of the integral from 0 to 3 of xe^xy).

Problem 3

Exercise 58 in section 15.2 of our textbook (find the volume of a solid between x² and the section of the xy plane bounded by the curves y = 2 - x² and y = x).

Problem 4

Exercise 86 in section 15.2 of our textbook, using muPad or a similar system (estimate the value of the integral from 0 to 1 of the integral from 0 to 1 of e^-(x2+y2).

Problem 5

Exercise 24 in section 15.3 of our textbook (find the area of a washer with outer radius 2 and inner radius 1, using double integration and simple geometry). Note that this is not a problem that demonstrates the value of double integration for finding areas, rather it applies the technique to a problem for which there are much simpler answers so that you can see that integration does produce the right answer. Consider using a table of integrals or muPad to evaluate some of the integrals you encounter in this problem. But note that I want exact answers, not numeric approximations, so simply numerically integrating the whole thing in muPad won’t serve.

Problem 6

In section 15.3, our book defines the integrals for calculating the area of a region and the average value of a function over a region as integrals with respect to area. In all of its examples of such things, however, the book integrates with respect to x and y. Is the book inconsistent in claiming that the integrals are with respect to area but then integrating with respect to x and y? If so, what is the problem? If not, why not?

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.